Secure computation system, secure computation apparatus, secure computation method, and recording medium

ABSTRACT

P i  and P +  have stored a + ∈{a 0 , a 1 , a 2 } and b + ∈{b 0 , b 1 , b 2 } therein, and P i  and P −  have stored a − ∈A −  and b − ∈B −  therein. Here, P + −P (i+1)mod 3 , P − =P (i−1)mod 3 , and a and b are arbitrary values and satisfy a=a 0 +a 1 +a 2  and b=b 0 +b 1 +b 2 , where A −  is a complement of a +  in {a 0 , a 1 , a 2 } and B −  is a complement of b +  in {b 0 , b 1 , b 2 }. P i  and P +  share r + , P i  and P −  share r − , and P i  calculates c + =(a + +a − )(b + +b − )−a − b − +r + −r − . P i  sends c +  to P + .

TECHNICAL FIELD

The present invention relates to the field of cryptographic techniques, and more particularly to the field of secure computation techniques.

BACKGROUND ART

In some schemes, values are exchanged among multiple secure computation apparatuses and each secure computation apparatus performs secure computation such as secure multiplication or secure multiply-accumulate using values received from other secure computation apparatuses (see Patent Literature 1, for instance).

PRIOR ART LITERATURE Patent Literature

Patent Literature 1: Japanese Registered Patent No. 6006842

SUMMARY OF THE INVENTION Problems to be Solved by the Invention

The scheme described in Patent Literature 1, however, has the disadvantage of involving a large number of additions/subtractions and multiplications and a large number of memory accesses.

An object of the present invention is to provide techniques for performing secure multiplication or secure multiply-accumulate with a less number of additions/subtractions and multiplications and a less number of memory accesses than conventional techniques.

Means to Solve the Problems

The secure multiplication according to the present invention is performed in the following manner. A storage of a secure computation apparatus P_(i) and a storage of a secure computation apparatus P₊ have stored therein subshares a₊∈{a₀, a₁, a₂} and b₊∈{b₀, b₁, b₂}, and the storage of the secure computation apparatus P_(i) and a storage of a secure computation apparatus P⁻ have stored therein subshares a⁻∈A⁻ and b⁻∈B⁻. Here, i∈{0, 1, 2} holds; P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a and b are arbitrary values; a₀, a₁, and a₂ are subshares satisfying a=a₀+a₁+a₂; b₀, b₁, and b₂ are subshares satisfying b=b₀+b₁+b₂; A⁻ is a complement of a₊ in {a₀, a₁, a₂}; and a is a complement of b₊ in {b₀, b₁, b₂}. A sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P₊ share an arbitrary value r₊, and the sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P⁻ share an arbitrary value r⁻, and an arithmetic unit of the secure computation apparatus P_(i) calculates c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻+r₊−r⁻. An output unit of each secure computation apparatus P_(i) outputs c₊ to the secure computation apparatus P₊, and c₊ is input to an input unit of the secure computation apparatus P₊.

The secure multiply-accumulate according to the present invention is performed in the following manner. A storage of a secure computation apparatus P_(i) and a storage of a secure computation apparatus P₊ have stored therein subshares a₊(n)∈{a₀(n), a₁(n), a₂(n)} and b₊(n)∈{b₀(n), b₁(n), b₂(n)}, and the storage of the secure computation apparatus P_(i) and a storage of a secure computation apparatus P⁻ have stored therein subshares a⁻(n)∈A⁻(n) and b⁻(n)∈B⁻(n). Here, i∈{0, 1, 2} holds; P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a(n) and b(n) are arbitrary values; a₀(n), a₁(n), and a₂(n) are subshares satisfying a(n)=a₀(n)+a₁(n)+a₂(n); b₀(n), b₁(n), and b₂(n) are subshares satisfying b(n)=b₀(n)+b₁(n)+b₂(n); N is a positive integer, where n=0, . . . , N−1; A⁻(n) is a complement of a₊(n) in {a₀(n), a₁(n), a₂(n)}; and B⁻(n) is a complement of b₊(n) in {b₀(n), b₁(n), b₂(n)}. A sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P₊ share an arbitrary value r₊, the sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P⁻ share an arbitrary value r⁻, and an arithmetic unit of the secure computation apparatus P_(i) calculates: c ₊=Σ_(n=0) ^(N-1){(a ₊(n)+a ⁻(n))(b ₊(n)+b ⁻(n))−a ⁻(n)b ⁻(n)+r ₊ −r ⁻}

Each secure computation apparatus P_(i) outputs c₊ to the secure computation apparatus P₊, and c₊ is input to an input unit of the secure computation apparatus P₊.

Effects of the Invention

This allows secure multiplication or secure multiply-accumulate to be performed with a less number of additions/subtractions and multiplications and a less number of memory accesses compared to conventional techniques.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating a secure computation system according to an embodiment.

FIG. 2 is a block diagram illustrating a secure computation apparatus according to the embodiment.

FIG. 3 is a conceptual diagram for explaining secure multiplication according to the embodiment.

FIG. 4 is a flow diagram for explaining a secure computation method according to the embodiment.

FIG. 5 is a conceptual diagram for explaining secure multiply-accumulate according to a further embodiment.

FIG. 6 is a flow diagram for explaining a secure computation method according to the further embodiment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Embodiments of the present invention are described below using the drawings.

First Embodiment

In a first embodiment, multiplication ab∈F of arbitrary values a, b∈F is performed by secure computation (secure multiplication). Here, F represents a finite field. This secure multiplication obtains secret sharing values of a multiplication result ab from secret sharing values (shares) of the arbitrary value a and secret sharing values of the arbitrary value b. This is described in detail below.

<Configuration>

As illustrated in FIG. 1, a secure computation system 1 of this embodiment includes three secure computation apparatuses 11-P₀, 11-P₁, 11-P₂ (secure computation apparatuses P₀, P₁, P₂). The secure computation apparatuses 11-P₀, 11-P₁, 11-P₂ are configured to be able to communicate over a communication network, such as the Internet. As illustrated in FIG. 2, a secure computation apparatus 11-P (where PE {P₀, P₁, P₂}) of this embodiment includes a storage 111-P (storage and memory), an arithmetic unit 112-P, a control unit 113-P, a sharing unit 114-P, and a communication unit 115-P. The secure computation apparatus 11-P executes various kinds of processing under control of the control unit 113-P and data resulting from the processing is sequentially stored in the storage 111-P and read out where necessary.

<Preliminary Processing>

Through preliminary processing, an arbitrary value a is secret-shared among subshares a₀, a₁, a₂∈F satisfying a=a₀+a₁+a₂∈F, and an arbitrary value b is secret-shared among subshares b₀, b₁, b₂∈F satisfying b=b₀+b₁+b₂∈F. A storage 111-P₁ of a secure computation apparatus 11-P_(i) has stored therein subshares a₊∈{a₀, a₁, a₂} and b₊∈{b₀, b₁, b₂}, and a storage 111-P₊ of a secure computation apparatus 11-P₊ also has stored therein the subshares a₊ and b₊. The subshare a₊ is any one element of {a₀, a₁, a₂}, and the subshare b₊ is any one element of {b₀, b₁, b₂}. The storage 111-P_(i) of the secure computation apparatus 11-P_(i) has stored therein subshares a⁻∈A⁻ and b⁻∈B⁻, and a storage 111-P⁻ of a secure computation apparatus 11-P⁻ also has stored therein subshares a⁻ and b⁻. The subshare a⁻ is any one element of A⁻, and the subshare b⁻ is any one element of B⁻. Here, A⁻ is a complement of a₊ in {a₀, a₁, a₂} (A⁻={a₀, a₁, a₂}|{a₊}), and B⁻ is a complement of b₊ in {b₀, b₁, b₂}(B⁻={b₀, b₁, b₂}−{b₊}). Also, i∈{0, 1, 2} holds, P₊ is P_((i+1)mod 3), and P⁻ is P_((i−1)mod 3). In the example of FIG. 3, subshares (a₀, a₁) and (b₀, b₁) are stored in the storage 111-P₀ of the secure computation apparatus 11-P₀, subshares (a₁, a₂) and (b₁, b₂) are stored in the storage 111-P₁ of the secure computation apparatus 11-P₁, and subshares (a₂, a₀) and (b₂, b₀) are stored in the storage 111-P₂ of the secure computation apparatus 11-P₂. Note that (a₀, a₁), (a₁, a₂), and (a₂, a₀) are secret sharing values of a, and (b₀, b₁), (b₁, b₂), and (b₂, b₀) are secret sharing values of b.

<Secure Computation>

On the precondition that this preliminary processing has been performed, each secure computation apparatus 11-P_(i) (where i∈{0, 1, 2}) performs the following secure computation (FIGS. 3 and 4).

A sharing unit 114-P_(i) of each secure computation apparatus 11-P_(i) and a sharing unit 114-P₊ of the secure computation apparatus 11-P₊ share an arbitrary value r₊∈F. That is, the sharing unit 114-P_(i) and the sharing unit 114-P₊ each obtain the same arbitrary value r₊ as that obtained by the other. The shared arbitrary value r₊ is stored in the storage 111-P_(i) and the storage 111-P₊. Examples of the arbitrary value r₊ include a pseudo random number, a true random number, a value selected from multiple predetermined values, an input value, a value resulting from a separate process, and the like. Sharing of the arbitrary value r₊ can be done in a known manner. For example, the arbitrary value r₊ may be shared between the sharing unit 114-P_(i) and the sharing unit 114-P₊ by sending the arbitrary value r₊ or information for identifying the arbitrary value r₊ from the sharing unit 114-P_(i) to the sharing unit 114-P₊. Conversely, the arbitrary value r₊ may be shared between the sharing unit 114-P_(i) and the sharing unit 114-P₊ by sending the arbitrary value r₊ or information for identifying the arbitrary value r₊ from the sharing unit 114-P₊ to the sharing unit 114-P_(i). The sharing unit 114-P_(i) and the sharing unit 114-P₊ may also share a seed value with each other beforehand and perform predetermined processing using the seed value so that the arbitrary value r₊ is shared between the sharing unit 114-P_(i) and the sharing unit 114-P₊. The arbitrary value r₊ may also be shared between the sharing unit 114-P_(i) and sharing unit 114-P₊ by means of a known key exchange algorithm (step S11).

The sharing unit 114-P_(i) of each secure computation apparatus 11-P_(i) and the sharing unit 114-P⁻ of the secure computation apparatus 11-P⁻ share an arbitrary value r⁻∈F. That is, the sharing unit 114-P_(i) and the sharing unit 114-P⁻ each obtain the same arbitrary value r⁻ as that obtained by the other. The shared arbitrary value r⁻ is stored in the storage 111-P_(i) and the storage 111-P⁻. Examples of the arbitrary value r⁻ include a pseudo random number, a true random number, a value selected from multiple predetermined values, an input value, a value resulting from a separate process, and the like. Sharing of the arbitrary value r⁻ can be done in a known manner. For example, the arbitrary value r⁻ may be shared between the sharing unit 114-P_(i) and the sharing unit 114-P⁻ by sending the arbitrary value r⁻ or information for identifying the arbitrary value r⁻ from the sharing unit 114-P_(i) to the sharing unit 114-P⁻. Conversely, the arbitrary value r⁻ may be shared between the sharing unit 114-P_(i) and the sharing unit 114-P⁻ by sending the arbitrary value r⁻ or information for identifying the arbitrary value r⁻ from the sharing unit 114-P⁻ to the sharing unit 114-P_(i). Alternatively, the sharing unit 114-P_(i) and the sharing unit 114-P⁻ may share a seed value with each other beforehand and perform predetermined processing using the seed value so that the arbitrary value r⁻ is shared between the sharing unit 114-P_(i) and the sharing unit 114-P⁻. The arbitrary value r⁻ may also be shared between the sharing unit 114-P_(i) and the sharing unit 114-P⁻ by means of a known key exchange algorithm (step S12).

In the example of FIG. 3, as the result of the processes at steps S11 and S12, the sharing unit 114-P₀ and the sharing unit 114-P₁ share an arbitrary value r₀₁∈F, the sharing unit 114-P_(i) and the sharing unit 114-P₂ share an arbitrary value r₁₂∈F, and the sharing unit 114-P₂ and the sharing unit 114-P₀ share an arbitrary value r₂₀∈F.

An arithmetic unit 112-P_(i) of each secure computation apparatus 11-P_(i) uses a₊, a⁻, b₊, b⁻, r₊, and r⁻ read from the storage 111-P_(i) to calculate and output c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻+r₊−r⁻∈F. In the example of FIG. 3, an arithmetic unit 112-P₀ of the secure computation apparatus 11-P₀ calculates and outputs c₁=(a₀+a₁)(b₀+b₁)−a₀b₀+r₀₁−r₂₀∈F, an arithmetic unit 112-P₁ of the secure computation apparatus 11-P₁ calculates and outputs c₂=(a₁+a₂)(b₁+b₂)−a₁b₁+r₁₂−r₀₁∈F, and an arithmetic unit 112-P₂ of the secure computation apparatus 11-P₂ calculates and outputs c₀=(a₂+a₀)(b₂+b₀)−a₂b₂+r₂₀−r₁₂∈F (step S13).

A communication unit 115-P_(i) of each secure computation apparatus 11-P₁ transmits c₊ (outputs c₊) to the secure computation apparatus 11-P₊ (step S14). c₊ is received by (input to) a communication unit 115-P₊ of the secure computation apparatus 11-P₊. Letting c⁻ represent a value calculated by an arithmetic unit 112-P⁻ of the secure computation apparatus 11-P⁻ and transmitted from a communication unit 115-P⁻, the communication unit 115-P_(i) of each secure computation apparatus 11-P_(i) receives c⁻ (step S15). In the example of FIG. 3, a communication unit 115-P₀ of the secure computation apparatus 11-P₀ transmits c₁ to the secure computation apparatus 11-P₁, and c₁ is received by a communication unit 115-P₁ of the secure computation apparatus 11-P₁. The communication unit 115-P₁ of the secure computation apparatus 11-P₁ transmits c₂ to the secure computation apparatus 11-P₂, and c₂ is received by a communication unit 115-P₂ of the secure computation apparatus 11-P₂. The communication unit 115-P₂ of the secure computation apparatus 11-P₂ transmits c₀ to the secure computation apparatus 11-P₀, and c₀ is received by the communication unit 115-P₀ of the secure computation apparatus 11-P₀ (steps S14 and S15).

c⁻ received at step S15 and c₊ obtained at step S13 are stored in the storage 111-P_(i) of each secure computation apparatus 11-P_(i). In the example of FIG. 3, (c₀, c₁) are stored in the storage 111-P₀ of the secure computation apparatus 11-P₀, (c₁, c₂) are stored in the storage 111-P_(i) of the secure computation apparatus 11-P₁, and (c₂, c₀) are stored in the storage 111-P₂ of the secure computation apparatus 11-P₂ (step S16).

c₀, c₁, and c₂ in this embodiment are the subshares of the multiplication result ab which satisfies ab=c₀+c₁+c₂∈F, and (c₀, c₁), (c₁, c₂), and (c₂, c₀) are the secret sharing values of the multiplication result ab. Obtaining any two of these secret sharing values (c₀, c₁), (c₁, c₂), (c₂, c₀) allows reconstruction of the multiplication result ab. That is, c₀+c₁+c₂=ab is satisfied. In the example of FIG. 3,

c₀ + c₁ + c₂ = (a₀ + a₁)(b₀ + b₁) − a₀b₀ + r₀₁ − r₂₀ + (a₁ + a₂)(b₁ + b₂) − a₁b₁ + r₁₂ − r₀₁ + (a₂ + a₀)(b₂ + b₀) − a₂b₂ + r₂₀ − r₁₂ = a₀b₁ + a₁b₀ + a₁b₁ + a₁b₂ + a₂b₁ + a₂b₂ + a₂b₀ + a₀b₂ + a₀b₀ = (a₀ + a₁ + a₂)(b₀ + b₁ + b₂) = ab  holds.

(c⁻, c₊) stored in the storage 111-P_(i) of each secure computation apparatus 11-P_(i) may be provided as input to a further secure computation apparatus (not shown), or may be provided as input to a reconstruction apparatus (not shown) which reconstructs the multiplication result ab for reconstruction and output of the multiplication result ab.

Features of this Embodiment

As shown above, this embodiment allows secure multiplication to be performed with a less number of additions/subtractions and multiplications and a less number of memory accesses compared to the conventional scheme described in Patent Literature 1. More specifically, the conventional scheme described in Patent Literature 1 requires each secure computation apparatus to save, in its memory, values obtained by performing additions/subtractions and multiplications on its own using secret sharing values or the like stored in the memory, to receive values that were obtained at another secure computation apparatus by additions/subtractions and multiplications and save them in the memory, and to further perform additions/subtractions and multiplications using these values saved in the memory. The scheme of Patent Literature 1 accordingly involves large numbers of additions/subtractions and multiplications and memory accesses. In this embodiment, by contrast, values that are obtained by each secure computation apparatus by performing additions/subtractions and multiplications on its own using secret sharing values and the like stored in its memory and values that are obtained at another secure computation apparatus by performing additions/subtractions and multiplication directly become the subshares of an arithmetic result. Thus, it involves a less number of additions/subtractions and multiplications and a less number of memory accesses than the scheme of Patent Literature 1. Particularly in secure computation of high arithmetic speed, the time for memory access creates a bottleneck in the improvement of the overall arithmetic speed. Because the scheme of this embodiment allows reduction in the number of memory accesses, it can significantly improve the arithmetic speed. For example, when calculating c_(XY)=a⁻b₊+a₊ b⁻−r_(ZX), secure multiplication with a conventional technique reads a⁻, b₊, a₊, and b⁻ and writes the resulting c_(XY). This requires execution of two multiplications and two additions/subtractions. Further, when calculating c⁻=a⁻b⁻+c_(ZX)+r_(ZX), secure multiplication with the conventional technique reads a⁻, b⁻, and c_(ZX) and writes c⁻. This requires execution of one multiplication and three additions/subtractions. Still further, when calculating c₊=a₊ b₊+c_(XY)+r_(XY), secure multiplication with the conventional technique reads a₊, b₊, and c_(XY) and writes the resulting c₊. This requires execution of one multiplication and three additions/subtractions. Consequently, in total, it is necessary to perform ten reading operations, three writing operations, four multiplications, and eight additions/subtractions per party. By contrast, when calculating c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻+r₊−r⁻, the secure computation apparatus 10-P_(i) of this embodiment reads a⁻, b₊, a₊, and b⁻ and writes the resulting c₊. Thus, in total, only four reading operations, one writing operation, two multiplications, and five additions/subtractions are required per party. Compared to the conventional technique, this embodiment can lessen the number of reading operations to 40%, the number of writing operations to 33%, the number of multiplications to 50%, and the number of additions/subtractions to 63%.

Second Embodiment

In a second embodiment, multiply-accumulate a(0)b(0)+ . . . +a(N−1)b(N−1)∈F with arbitrary values a(0), a(N−1) and b(0), b(N−1) is performed by secure computation (secure multiply-accumulate). Here, F represents a finite field and N is a positive integer (for example, an integer greater than 1). This secure multiply-accumulate obtains secret sharing values of a multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1) from secret sharing values of an arbitrary value a(n) and secret sharing values of an arbitrary value b(n) for n=0, . . . , N−1. This is described in detail below.

<Configuration>

As illustrated in FIG. 1, a secure computation system 2 of this embodiment includes three secure computation apparatuses 21-P₀, 21-P₁, 21-P₂ (secure computation apparatuses P₀, P₁, P₂). The secure computation apparatuses 21-P₀, 21-P₁, 21-P₂ are configured to be able to communicate over a communication network, such as the Internet. As illustrated in FIG. 2, a secure computation apparatus 21-P (where P∈{P₀, P₁, P₂}) of this embodiment includes a storage 211-P (storage and memory), an arithmetic unit 212-P_(i) a control unit 213-P_(i) a sharing unit 114-P_(i) and a communication unit 115-P. The secure computation apparatus 21-P executes various kinds of processing under control of the control unit 213-P and data resulting from the processing is sequentially stored in the storage 211-P and read out where necessary.

<Preliminary Processing>

Through preliminary processing, an arbitrary value a(n) is secret-shared among subshares a₀(n), a₁(n), a₂(n)∈F satisfying a(n)=a₀(n)+a₁(n)+a₂(n)∈F, and an arbitrary value b(n) is secret-shared among subshares b₀(n), b₁(n), b₂(n)∈F satisfying b(n)=b₀(n)+b₁(n)+b₂(n)∈F, for n=0, . . . , N−1 (where N is a positive integer, for example, N is an integer greater than 1). A storage 211-P_(i) of a secure computation apparatus 21-P_(i) has stored therein subshares a₊(n)∈{a₀(n), a₁(n), a₂(n)} and b₊(n)∈{b₀(n), b₁(n), b₂(n)} for n=0, . . . , N−1, and a storage 211-P₊ of a secure computation apparatus 21-P₊ also has stored therein the subshares a₊(n) and 13 ₊(n) for n=0, . . . , N−1. Each subshare a₊(n) is any one element of {a₀(n), a₁(n), a₂(n)}, and each subshare b₊(n) is any one element of {b₀(n), b₁(n), b₂(n)}. The storage 211-P_(i) of the secure computation apparatus 21-P_(i) has stored therein subshares a⁻(n)∈A⁻(n) and b⁻(n)∈B⁻(n) for n=0, . . . , N−1, and a storage 211-P⁻ of a secure computation apparatus 21-P⁻ also has stored therein subshares a⁻(n) and b⁻(n) for n=0, . . . , N−1. Each subshare a⁻(n) is any one element of A⁻(n), and each subshare b⁻(n) is any one element of B⁻(n). Here, A⁻(n) is a complement of a₊(n) in {a₀(n), a₁(n), a₂(n)}(A⁻(n)={a₀(n), a₁(n), a₂(n)}-{a₊(n)}), and B⁻(n) is a complement of b₊(n) in {b₀(n), b₁(n), b₂(n)}(B⁻(n)={b₀(n), b₁(n), b₂(n)}-{b₊(n)}). Also, i∈{0, 1, 2} holds, P₊ is P_((i+1)mod 3), and P⁻ is P_((i−1)mod 3). In the example of FIG. 5, subshares (a₀(n), a₁(n)) and (b₀(n), b₁(n)) for n=0, . . . , N−1 are stored in a storage 211-P₀ of the secure computation apparatus 21-P₀, subshares (a₁(n), a₂(n)) and (b₁(n), b₂(n)) for n=0, . . . , N−1 are stored in a storage 211-P_(i) of the secure computation apparatus 21-P₁, and subshares (a₂(n), a₀(n)) and (b₂(n), b₀(n)) for n=0, . . . , N−1 are stored in a storage 211-P₂ of the secure computation apparatus 21-P₂. Note that (a₀(n), a₁(n)), (a₁(n), a₂(n)), and (a₂(n), a₀(n)) are secret sharing values of a(n), and (b₀(n), b₁(n)), (b₁(n), b₂(n)), and (b₂(n), b₀(n)) are secret sharing values of b(n).

<Secure Computation>

On the precondition that this preliminary processing has been performed, each secure computation apparatus 21-P_(i) (where i∈{0, 1, 2}) performs the following secure computation (FIGS. 5 and 6).

A sharing unit 114-P_(i) of each secure computation apparatus 21-P_(i) and a sharing unit 114-P₊ of the secure computation apparatus 21-P₊ share an arbitrary value r₊ E F. The shared arbitrary value r₊ is stored in the storage 211-P_(i) and the storage 211-P₊. Specific examples of the arbitrary value r₊ and the process of sharing it were described in the first embodiment (step S21).

The sharing unit 114-P_(i) of each secure computation apparatus 21-P_(i) and the sharing unit 114-P⁻ of the secure computation apparatus 21-P⁻ share an arbitrary value r⁻∈F. The shared arbitrary value r⁻ is stored in the storage 211-P_(i) and the storage 211-P⁻. Specific examples of the arbitrary value r⁻ and the process of sharing it were described in the first embodiment (step S22).

In the example of FIG. 5, as the result of the processes at steps S21 and S22, the sharing unit 114-P₀ and the sharing unit 114-P₁ share an arbitrary value r₀₁∈F, the sharing unit 114-P₁ and the sharing unit 114-P₂ share an arbitrary value r₁₂∈F, and the sharing unit 114-P₂ and the sharing unit 114-P₀ share an arbitrary value r₂₀∈F.

An arithmetic unit 212-P_(i) of each secure computation apparatus 21-P_(i) uses a₊(n), a⁻(n), b₊(n), b⁻(n), r₊, and r⁻ for n=0, . . . , N−1 read from the storage 211-P_(i) to calculate and output: c ₊=Σ_(n=0) ^(N-1){(a ₊(n))+a ⁻(n))(b ₊(n)+b ⁻(n))−a ⁻(n)b ⁻(n)+r ₊ −r ⁻ }∈F

In the example of FIG. 5, the arithmetic unit 212-P₀ of the secure computation apparatus 21-P₀ calculates and outputs: c ₁=Σ_(n=0) ^(N-1){(a ₀(n))+a ₁(n))(b ₀(n)+b ₁(n))−a ₀(n)b ₀(n)+r ₀₁ −r ₂₀ }∈F the arithmetic unit 212-P₁ of the secure computation apparatus 21-P_(i) calculates and outputs: c ₂=Σ_(n=0) ^(N-1){(a ₁(n))+a ₂(n))(b ₁(n)+b ₂(n))−a ₁(n)b ₁(n)+r ₁₂ −r ₀₁ }∈F and the arithmetic unit 212-P₂ of the secure computation apparatus 21-P₂ calculates and outputs: c ₀=Σ_(n=0) ^(N-1){(a ₂(n))+a ₀(n))(b ₂(n)+b ₀(n))−a ₂(n)b ₂(n)+r ₂₀ −r ₁₂ }∈F (step S23).

A communication unit 115-P_(i) of each secure computation apparatus 21-P_(i) transmits c₊ (outputs c₊) to the secure computation apparatus 21-P₊ (step S24). c₊ is received by (input to) a communication unit 115-P₊ of the secure computation apparatus 21-P₊. Letting c⁻ represent a value calculated by an arithmetic unit 212-P⁻ of the secure computation apparatus 21-P⁻ and transmitted from the communication unit 115-P⁻ the communication unit 115-P_(i) of each secure computation apparatus 21-P_(i) receives c⁻ (step S25). In the example of FIG. 5, a communication unit 115-P₀ of the secure computation apparatus 21-P₀ transmits c₁ to the secure computation apparatus 21-P₁, and c₁ is received by a communication unit 115-P_(i) of the secure computation apparatus 21-P₁. The communication unit 115-P₁ of the secure computation apparatus 21-P₁ transmits c₂ to the secure computation apparatus 21-P₂, and c₂ is received by a communication unit 115-P₂ of the secure computation apparatus 21-P₂. The communication unit 115-P₂ of the secure computation apparatus 21-P₂ transmits c₀ to the secure computation apparatus 21-P₀, and c₀ is received by the communication unit 115-P₀ of the secure computation apparatus 21-P₀ (steps S24 and S25).

c⁻ received at step S25 and c₊ obtained at step S23 are stored in the storage 211-P_(i) of each secure computation apparatus 21-P_(i). In the example of FIG. 5, (c₀, c₁) are stored in the storage 211-P₀ of the secure computation apparatus 21-P₀, (c₁, c₂) are stored in the storage 211-P₁ of the secure computation apparatus 21-P₁, and (c₂, c₀) are stored in the storage 211-P₂ of the secure computation apparatus 21-P₂ (step S16).

c₀, c₁, and c₂ in this embodiment are the subshares of the multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1), which satisfies a(0)b(0)+ . . . +a(N−1)b(N−1)=c₀+c₁+c₂∈F. (c₀, c₁), (c₁, c₂), and (c₂, c₀) are the secret sharing values of the multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1). Obtaining any two of these secret sharing values (c₀, c₁), (c₁, c₂), (c₂, c₀) allows reconstruction of the multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1). That is, c₀+c₁+c₂=a(0)b(0)+ . . . +a(N−1)b(N−1) is satisfied.

(c⁻, c₊) stored in the storage 211-P_(i) of each secure computation apparatus 21-P_(i) may be provided as input to a further secure computation apparatus (not shown), or may be provided as input to a reconstruction apparatus (not shown) which reconstructs the multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1) for reconstruction and output of the multiply-accumulate result a(0)b(0)+ . . . +a(N−1)b(N−1).

Features of this Embodiment

As shown above, this embodiment allows secure multiply-accumulate to be performed with a less number of additions/subtractions and multiplications and a less number of memory accesses compared to the conventional scheme described in Patent Literature 1.

It is to be noted that the present invention is not limited to the foregoing embodiments. For example, the above-described various kinds of processing may be executed, in addition to being executed in chronological order in accordance with the descriptions, in parallel or individually depending on the processing power of an apparatus that executes the processing or when needed. In addition, it goes without saying that changes may be made as appropriate without departing from the spirit of the present invention. Also, the arbitrary value r₊ in the second embodiment may be r₊(n) corresponding to each one of n=0, . . . , N−1, and the arbitrary value r⁻ may be r⁻(n) corresponding to each one of n=0, . . . , N−1. For instance, in the example of FIG. 5, the arbitrary value r₀₁ may be r₀₁(n)∈F corresponding to each one of n=0, . . . , N−1, and the arbitrary value r₂₀ may be r₂₀(n)∈F corresponding to each one of n=0, . . . , N−1.

The above-described each apparatus is embodied by execution of a predetermined program by a general- or special-purpose computer having a processor (hardware processor) such as a central processing unit (CPU), memories such as random-access memory (RAM) and read-only memory (ROM), and the like, for example. The computer may have one processor and one memory or have multiple processors and memories. The program may be installed on the computer or pre-recorded on the ROM and the like. Also, some or all of the processing units may be embodied using an electronic circuit that implements processing functions without using programs, rather than an electronic circuit (circuitry) that implements functional components by loading of programs like a CPU. An electronic circuit constituting a single apparatus may include multiple CPUs.

When the above-described configurations are implemented by a computer, the processing details of the functions supposed to be provided in each apparatus are described by a program. As a result of this program being executed by the computer, the above-described processing functions are implemented on the computer. The program describing the processing details can be recorded on a computer-readable recording medium. An example of the computer-readable recording medium is a non-transitory recording medium. Examples of such a recording medium include a magnetic recording device, an optical disk, a magneto-optical recording medium, and semiconductor memory.

The distribution of this program is performed by, for example, selling, transferring, or lending a portable recording medium such as a DVD or a CD-ROM on which the program is recorded. Furthermore, a configuration may be adopted in which this program is distributed by storing the program in a storage device of a server computer and transferring the program to other computers from the server computer via a network.

The computer that executes such a program first, for example, temporarily stores the program recorded on the portable recording medium or the program transferred from the server computer in a storage device thereof. At the time of execution of processing, the computer reads the program stored in the storage device thereof and executes the processing in accordance with the read program. As another mode of execution of this program, the computer may read the program directly from the portable recording medium and execute the processing in accordance with the program and, furthermore, every time the program is transferred to the computer from the server computer, the computer may sequentially execute the processing in accordance with the received program. A configuration may be adopted in which the transfer of a program to the computer from the server computer is not performed and the above-described processing is executed by so-called application service provider (ASP)-type service by which the processing functions are implemented only by an instruction for execution thereof and result acquisition.

Instead of executing a predetermined program on the computer to implement the processing functions of the present apparatuses, at least some of the processing functions may be implemented by hardware.

DESCRIPTION OF REFERENCE NUMERALS

1, 2 secure computation system

11-P secure computation apparatus 

What is claimed is:
 1. A secure computation system comprising: a secure computation apparatus P₀; a secure computation apparatus P_(i); and a secure computation apparatus P₂, wherein each of the secure computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a and b, wherein the data a and b is concealed from each of the secure computation apparatuses; i∈{0, 1, 2} holds; P₊ is P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a and b are arbitrary values; a₀, a₁, and a₂ are subshares satisfying a=a₀+a₁+a₂; b₀, b₁, and b₂ are subshares satisfying b=b₀+b₁+b₂, a storage of a secure computation apparatus P_(i) and a storage of a secure computation apparatus P₊ have stored therein subshares a₊∈{a₀, a₁, a₂} and b₊∈{b₀, b₁, b₂}, the storage of the secure computation apparatus P_(i) and a storage of a secure computation apparatus P⁻ have stored therein subshares a⁻∈A⁻ and b⁻∈B⁻, where A⁻ is a complement of a₊ in {a₀, a₁, a₂} and B⁻ is a complement of b₊ in {b₀, b₁, b₂}, a sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P₊ share an arbitrary value r₊, wherein r₊∈F, the sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P⁻ share an arbitrary value r⁻, wherein r⁻∈F, an arithmetic unit of the secure computation apparatus P_(i) calculates c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻+r₊−r⁻, wherein c₊∈{c₀, c₁, c₂}, and c₀+c₁+c₂=a*b, and an output unit of the secure computation apparatus P_(i) outputs c₊ to the secure computation apparatus P₊, and c₊ is input to an input unit of the secure computation apparatus P₊, which results in an exchange process of secure computation apparatus P₀ calculating c_(i) and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating and transmitting c₂ to secure computation apparatus P₂ and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₂.
 2. A secure computation system comprising: a secure computation apparatus P₀; a secure computation apparatus P₁; and a secure computation apparatus P₂, wherein each of the secret computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a(n) and b(n), wherein the data a(n) and b(n) is concealed from each of the secret computation apparatuses; i∈{0, 1, 2} holds; P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a(n) and b(n) are arbitrary values; a₀(n), a₁(n), and a₂(n) are subshares of a(n) satisfying a(n)=a₀(n)+a₁(n)+a₂(n); b₀(n), b₁(n), and b₂(n) are subshares of b(n) satisfying b(n)=b₀(n)+b₁(n)+b₂(n); and N is a positive integer, where n=0, . . . , N−1, a storage of a secure computation apparatus P_(i) and a storage of a secure computation apparatus P₊ have stored therein subshares a₊(n)∈{a₀(n), a₁(n), a₂(n)} and b₊(n)∈{b₀(n), b₁(n), b₂(n)}, the storage of the secure computation apparatus P_(i) and a storage of a secure computation apparatus P⁻ have stored therein subshares a⁻(n)∈A⁻(n) and b⁻(n)∈B⁻(n), where A⁻(n) is a complement of a₊(n) in {a₀(n), a₁(n), a₂(n)} and B⁻(n) is a complement of b₊(n) in {b₀(n), b₁(n), b₂(n)}, a sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P₊ share an arbitrary value r₊, wherein r₊∈F, the sharing unit of the secure computation apparatus P_(i) and a sharing unit of the secure computation apparatus P⁻ share an arbitrary value r⁻, wherein r⁻∈F, an arithmetic unit of the secure computation apparatus P_(i) calculates: c ₊=Σ_(n=0) ^(N-1){(a ₊(n)+a ⁻(n))(b ₊(n)+b ⁻(n))−a ⁻(n)b ⁻(n)+r ₊ −r ⁻}, wherein c₊{c₀, c₁, c₂}, and c₀+c₁+c₂=a(0)b(0)+ . . . +a(N−1)b(N−1), and an output unit of the secure computation apparatus P_(i) outputs c₊ to the secure computation apparatus P₊, and c₊ is input to an input unit of the secure computation apparatus P₊, which results in an exchange process of secure computation apparatus P₀ calculating c₁ and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating c₂ and transmitting c₂ to secure computation apparatus P₂ and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₇.
 3. A secure computation apparatus, wherein the secure computation apparatus is one of a plurality of secure computation apparatuses P₀, P₁, and P₂, each of the secure computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a and b, wherein the data a and b is concealed from each of the secure computation apparatuses, i∈{0, 1, 2} holds; P₊ is P_((i−1)mod 3), P⁻ is P_((i−1)mod 3), a and b are arbitrary values; a₀, a₁, and a₂ are subshares of a satisfying a=a₀+a₁+a₂, and b₀, b₁, and b₂ are subshares of b satisfying b=b₀+b₁+b₂, and the secure computation apparatus includes a storage that stores subshares a₊∈{a₀, a₁, a₂} and b₊∈{b₀, b₁, b₂} which are stored on a secure computation apparatus P and subshares a⁻∈A⁻ and b⁻∈B⁻ which are stored on a secure computation apparatus P⁻ (where A⁻ is a complement of a₊ in {a₀, a₁, a₂} and B⁻ is a complement of b₊ in {b₀, b₁, b₂}), a sharing unit that shares an arbitrary value r₊ with the secure computation apparatus P₊ and shares an arbitrary value r⁻ with the secure computation apparatus P⁻, wherein r₊∈F and r⁻∈F, an arithmetic unit that calculates c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻r₊r⁻, wherein c₊∈{c₀, c₁, c₂}, and c₀+c₁+c₂=a*b, and an output unit that outputs c₊ to the secure computation apparatus P₊, which results in an exchange process of secure computation apparatus P₀ calculating c₁ and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating c₂ and transmitting c₂ to secure computation apparatus P₂ and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₂.
 4. A secure computation apparatus, wherein the secure computation apparatus is one of a plurality of secure computation apparatuses P₀, P₁, and P₂, each of the secure computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a(n) and b(n), wherein the data a(n) and b(n) is concealed from each of the secure computation apparatuses; i∈{0, 1, 2} holds; P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a(n) and b(n) are arbitrary values; a₀(n), a₁(n), and a₂(n) are subshares of a(n) satisfying a(n)=a₀(n)+a₁(n)+a₂(n); and b₀(n), b₁(n), and b₂(n) are subshares of b(n) satisfying b(n)=b₀(n)+b₁(n)+b₂(n), and the secure computation apparatus includes a storage that stores subshares a₊(n)∈{a₀(n), a₁(n), a₂(n)} and b₊(n)∈{b₀(n), b₁(n), b₂(n)} which are stored on a secure computation apparatus P₊, and subshares a⁻(n)∈A⁻(n) and b⁻(n)∈B⁻(n) which are stored on a secure computation apparatus P⁻ (where A⁻(n) is a complement of a₊(n) in {a₀(n), a₁(n), a₂(n)} and B⁻(n) is a complement of b₊(n) in {b₀(n), b₁(n), b₂(n)}), a sharing unit that shares an arbitrary value r₊ with the secure computation apparatus P₊ and shares an arbitrary value r⁻ with the secure computation apparatus P⁻, wherein r₊∈F and r⁻∈F, an arithmetic unit that calculates: c ₊=Σ_(n=0) ^(N-1){(a ₊(n)+a ⁻(n))(b ₊(n)+b ⁻(n))−a ⁻(n)b ⁻(n)+r ₊ −r ⁻}, wherein c₊∈{c₀, c₁, c₂}, and c₀+c₁+c₂=a(0)b(0)+ . . . +a(N−1)b(N−1), and an output unit that outputs to the secure computation apparatus P₊₊, which results in an exchange process of secure computation apparatus P₀ calculating c₁ and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating c₂ and transmitting c₂ to secure computation apparatus P₂ and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₇.
 5. The secure computation apparatus according to claim 4, wherein the arbitrary value r₊ is r₊(n) and the arbitrary value r⁻ is r⁻(n).
 6. A computer-readable recording medium storing a program for causing a computer to function as the secure computation apparatus according to claim 3 or
 4. 7. A secure computation method implemented by a secure computation apparatus, wherein the secure computation apparatus is one of a plurality of secure computation apparatuses P₀, P₁, and P₂, each of the secure computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a and b, wherein the data a and b is concealed from each of the secure computation apparatuses; i∈{0, 1, 2} holds; P₊ is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a and b are arbitrary values; a₀, a₁, and a₂ are subshares of a satisfying a=a₀+a₁+a₂; and b₀, b₁, and b₂ are subshares of b satisfying b=b₀+b₁+b₂, subshares a₊∈{a₀, a₁, a₂} and b₊∈{b₀, b₁, b₂} stored on a secure computation apparatus P₊, and subshares a⁻∈A⁻ and b⁻∈B⁻ stored on a secure computation apparatus P⁻ are stored in a storage of a secure computation apparatus P_(i), where A⁻ is a complement of a₊ in {a₀, a₁, a₂} and B⁻ is a complement of b₊ in {b₀, b₁, b₂}, and the secure computation method includes a step of sharing, by a sharing unit of the secure computation apparatus P_(i), an arbitrary value r₊ with the secure computation apparatus P₊ and sharing an arbitrary value r⁻ with the secure computation apparatus P⁻, wherein r₊∈F and r⁻∈F, a step of calculating c₊=(a₊+a⁻)(b₊+b⁻)−a⁻b⁻+r₊−r⁻ by an arithmetic unit of the secure computation apparatus P_(i), wherein c₊∈{c₀, c₁ c₂}, and c₀+c₁+c₂=a*b, and a step of outputting c₊ to the secure computation apparatus P₊ by an output unit of the secure computation apparatus P_(i+), which results in an exchange process of secure computation apparatus P₀ calculating c₁ and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating c₂ and transmitting c₂ to secure computation apparatus P₂, and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₂.
 8. A secure computation method, implemented by a secure computation apparatus, wherein the secure computation apparatus is one of a plurality of secure computation apparatuses P₀, P₁, and P₂, each of the secure computation apparatuses comprising processing circuitry configured to receive, over a network, an input of one or more respective subshares of data a(n) and b(n), wherein the data a(n) and b(n) is concealed from each of the secure computation apparatuses; i∈{0, 1, 2} holds; P is P_((i+1)mod 3); P⁻ is P_((i−1)mod 3); a(n) and b(n) are arbitrary values; a₀(n), a₁ (n), and a₂(n) are subshares of a(n) satisfying a(n)=a₀(n)+a₁(n)+a₂(n); and b₀(n), b₁(n), and b₂(n) are subshares of b(n) satisfying b(n)=b₀(n)+b₁(n)+b₂(n), subshares a₊(n)∈{a₀(n), a₁(n), a₂(n)} and b₊(n)∈{b₀(n), b₁(n), b₂(n)} stored on a secure computation apparatus P₊, and subshares a⁻(n)∈A⁻(n) and b⁻(n)∈B⁻(n) stored on a secure computation apparatus P⁻ are stored in a storage of a secure computation apparatus P_(i), where A⁻(n) is a complement of a₊(n) in {a₀(n), a₁(n), a₂(n)} and B⁻(n) is a complement of b₊(n) in {b₀(n), b₁(n), b₂(n)}, and the secure computation method includes a step of sharing, by a sharing unit of the secure computation apparatus P_(i), an arbitrary value r₊ with the secure computation apparatus P₊ and sharing an arbitrary value r⁻ with the secure computation apparatus P⁻, wherein r₊∈F and r⁻∈F, a step of calculating, by an arithmetic unit of the secure computation apparatus P_(i): c ₊=Σ_(n=0) ^(N-1){(a ₊(n)+a ⁻(n))(b ₊(n)+b ⁻(n))−a ⁻(n)b ⁻(n)+r ₊ −r ⁻}, wherein c₊∈{c₀, c₁, c₂}, and c₀+c₁+c₂=a(0)b(0)+ . . . +a(N−1)b(N−1), and a step of outputting to the secure computation apparatus P₊ by an output unit of the secure computation apparatus P_(i), which results in an exchange process of secure computation apparatus P₀ calculating c₁ and transmitting c₁ to secure computation apparatus P₁, secure computation apparatus P₁ calculating c₂ and transmitting c₂ to secure computation apparatus P₂, and secure computation apparatus P₂ calculating c₀ and transmitting c₀ to secure computation apparatus P₀, wherein after the exchange process, each of P₀, P₁, and P₂ store a different pair of values among c₀, c₁, c₂. 